Let $f$ be a twice differentiable function, and let $f(-7)=6$, $f'(-7)=0$, and $f''(-7)=-5$. What occurs in the graph of $f$ at the point $(-7,6)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-7,6)$ is a minimum point. (Choice B) B $(-7,6)$ is a maximum point. (Choice C) C There's not enough information to tell.
Explanation: Since $f'(-7)=0$, we know that $x=-7$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $f$ at this point according to these three cases: If $f''(-7)>0$, the graph of $f$ has a minimum point at $x=-7$. If $f''(-7)<0$, the graph of $f$ has a maximum point at $x=-7$. If $f''(-7)=0$, the test is inconclusive. [Why is this so?] We are given that $f''(-7)=-5<0$. Therefore, $(-7,6)$ is a maximum point.